Protein folding reaction

The folding of protein is a first-order reaction because only one molecule is involved at a time. The reaction rates then became $\begin{aligned} \frac{d [A]}{d t} &= - k_{A \to B} [A] + k_{B \to A} [B] \\ \frac{d [B]}{d t} &= - k_{B \to A} [B] + k_{A \to B} [A] \end{aligned}$ where the coefficients, $k_{A \to B}$ and $k_{B \to A}$ , describe the foward and backward reaction, respectively. In equilibrium, the net reaction rates are zero, so that $\frac{d [A]}{d t} = \frac{d [B]}{d t} = 0 \quad \Rightarrow \quad \frac{[B]}{[A]} = \frac{k_{A \to B}}{k_{B \to A}}$ The coefficients follow Arhenius law $\begin{aligned} k_{A \to B} &= C \exp\left( - \frac{\Delta G_A }{R T}\right)\\ k_{B \to A} &= C \exp\left( - \frac{\Delta G_A - \Delta G}{R T}\right) \end{aligned}$ where $\Delta G_A$ is the activation free enthalpy and $\Delta G < 0$ is the reaction free enthalpy.

Finally, the concentration ratio results to $\frac{[B]}{[A]} = \frac{k_{A \to B}}{k_{B \to A}} = \exp\left( - \frac{ \Delta G}{R T}\right) = \exp\left( \frac{23000 }{8.314 \cdot 298}\right) \approx 1.08 \cdot 10^4$